### arXiv:1201.4920 [math.AP]AbstractReferencesReviewsResources

#### Traveling Wave Solutions of Advection-Diffusion Equations with Nonlinear Diffusion

Léonard Monsaingeon, Alexeï Novikov, Jean-Michel Roquejoffre

Published 2012-01-24Version 1

We study the existence of particular traveling wave solutions of a nonlinear parabolic degenerate diffusion equation with a shear flow. Under some assumptions we prove that such solutions exist at least for propagation speeds c {\in}]c*, +{\infty}, where c* > 0 is explicitly computed but may not be optimal. We also prove that a free boundary hy- persurface separates a region where u = 0 and a region where u > 0, and that this free boundary can be globally parametrized as a Lipschitz continuous graph under some additional non-degeneracy hypothesis; we investigate solutions which are, in the region u > 0, planar and linear at infinity in the propagation direction, with slope equal to the propagation speed.

**Comments:**40 pages, 1 figure

**Categories:**math.AP